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Metamath Proof Explorer

Theorem stoic3

Description: Stoic logic Thema 3. Statement T3 of Bobzien p. 116-117 discusses Stoic logic Thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external assumption, another follows, then that other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019)

	Ref	Expression
Hypotheses	stoic3.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
	stoic3.2	⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion	stoic3	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )

Proof

Step	Hyp	Ref	Expression
1		stoic3.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		stoic3.2	⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜏 )
3	1 2	sylan	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜃 ) → 𝜏 )
4	3	3impa	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜃 ) → 𝜏 )